Chapter 3: Differentiation PDF

112 Chapter 3: Differentiation Now take limits as h --+ o. By Theorem I of Section 2.2, lim j(c + h) = lim j(c) + lim j(c + h) -...

112 Chapter 3: Differentiation Now take limits as h --+ o. By Theorem I of Section 2.2, lim j(c + h) = lim j(c) + lim j(c + h) - j(c) lim h 11-0 11-0 11-0 h 11-0 = j(c) + f'(c)' 0 = j(c) + 0 = j(c). Similar arguments with one-sided limits show that if j has a derivative from one side (right or left) at x = c then j is continuous from that side at x = c. Theorem I says that if a function has a discontinuity at a point (for instance, a jump discontinuity), then it cannot be differentiable there. The greatest integer function Y = l x J fails to be differentiable at every integer x = n (Example 4, Section 2.5). Caution The converse of Theorem I is false. A function need not have a derivative at a point where it is continuous, as we saw in Example 4. Exercises 3.2 Finding Derivative Functions and values In Exercises 17-18, differentiate the functions. Then Imd an equation Using the delmition, calculate the derivatives of the functions in Exer- of the tangent line at the indicated point on the graph of the function. cises I-J). Then Imd the values of the derivatives as specified. 8 1. j(x) = 4 - x 2 ; /,(-3), /,(0), /,(1) 17. y = j(x) =.'------;:' (x,y) = (6,4) yx - 2 2. F(x) = (x - 1)2 + I; F'(-I),F'(0),F'(2) 18. w = g(z) = I +~, (z, w) = (3,2) 3. g(l) = 1,; 1 g'( -I), g'(2), g'( \13) In Exercises 1!)"22, find the values of the derivatives. ds 19. l ifs=I-312 4. k(z) = I ; z; k'( -I), k'(i), k'( Yz) dt 1=-1 5. p(O) = v30; p'(1),p'(3),p'(2/3) 20. dyl if y= I-~ 6. r(s) = ~; r'(O), r'(I), r'(1/2) "" x-v. drl if r = 2 In Exercises 7-12, Imd the indicated derivatives. 21. dO '-0 v'4=9 7.""dy ,· f' y=2s 1 8.""dr ,·f r = s' - 29 2 I +3 22. :t. if w=z+ Yz "" 9. dl if s=21+1 10 dv dt if v = t - t Using the Alternative Formula for Derivatives dp I dz I 11. dq if p= 12. dw if z= Use the formula yq:tJ Y/3w - 2 Slopes and rangent Lines /,(x) = lim j(z) - j(x) In Exercises 13-16, differentiate the functions aod Imd the slope of %-x Z X the tangent line at the given value of the independent variable. to Imd the derivative of the functions in Exercises 23-26. 9 13. j(x) = x + x' x = -3 I 23. j(x) = x + 2 I 14. k(x) = 2 + x' x = 2 24. j(x) = x 2 - 3x +4 15. s = I' - 12, 1 = -I x 25. g(x) = - - I x- x+3 16. y = -I- , -x x = -2 26. g(x) = I + Vx  3.2 The Derivative as a Function 113 Graphs 32. Recovering a function from its derivative Match the functions graphed in Exercises 27-30 with the derivatives a. Use the following information to graph the function f over graphed in the accompanying figures (a)-(d). the closed interval [ - 2, 5]. i) The graph off is made of closed line segments joined end to end. ii) The graph starts at the point ( - 2, 3). iii) The derivative of / is the step function in the figore shown here. y' (a) (b) 1 0-----0 y' C>--_ - -1- -2 b. Repeat part (a) assuming that the graph starts at (-2,0) (c) (d) instead of( -2,3). 33. Growth in the economy The graph in the accompanying figore shows the average aonual percentage change y ~ /(t) in the u.s. 27. y 28. y gross natioua1 product (GNP) for the years 1983-1988. Graph dy/dt (where defmed). 7% 6 --fr\ ------~~~----~x o ------~~~----~x 5 4 3 I- l \............. /" - 2 1 29. y 30. y o 1983 1984 1985 1986 1987 1988 34. Fruit rues (Continuation of Example 4, Section 2.1.) Popula- tions starting out in closed environments grow slowly at fust, ---\--~~4---+-~x when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are still abundant, then slowly again as the population reaches the carrying capacity of the environment a, Use the graphical tecbrtique of Example 3 to graph the deriva- 31. a. The graph in the accompanying figore is made of line seg- tive of the fruit fly population. The graph of the population is ments joined end to eml. At which points of the interval reproduced here. [-4, 6] is f' not defined? Give reasons for your answer. p y 350 300 ,..... V ~ 250 'a 200 / Loo Z 150 50 / /./ (1. -2) (4.-2) o 10 20 30 40 50 TIme (days) b. Graph the derivative off. b. During what dsys does the population seem to be increasing The graph should show a step function. fastest? Slowest? 122 Chapter 3: Differentiation I How to Read the Symbols for If y" is differentiable, its derivative, y'" = dy" /dx = d'y/dx', is the third derivative ofy with respect to x. The names continue as you imagine, with Derivatives y' ''y prime" d d"y y. 'y double prime" y(") = dxy("-l) = dx" = D"y d'y "d squared y dx squared" dx' denoting the nth derivative of y with respect to x for any positive integer n. y~ ''y triple prime" We can interpret the second derivative as the rate of change of the slope of the tangent yv.) ''y super n" to the graph of y = f{x) at each point. You will see in the next chapter that the second de- d"y rivative reveals whether the graph bends upward or downward from the tangent line as we "d to the n of y by dx to the n" dx" move off the point of tangency. In the next section, we interpret both the second and third D" "Dtothen" derivatives in terms of motion along a straight line. EXAMPLE 8 The flISt four derivatives of y = x' - 3x' + 2 are First derivative: y' = 3x' - 6x Second derivative: y" = 6x - 6 Third derivative: y.' = 6 Fourth derivative: y(4) = O. The function has derivatives of all orders, the fifth and later derivatives all being zero. _ Exercises 3.3 Derivative Calculations = I + x - 4v'i In Exercises 1-12, fmd the fITSt and second derivatives. 25. v x 1. y = -x' + 3 2. Y = x' + x + 8 I (x+I)(x+2) 3 = 51' - 31' 4. w = 3z' - 7z 3 + 21z2 27. Y = 28. Y = ~---c~-~ (x' - I)(x' + x + I) (x - I)(x - 2) 4x' x3 x2 x 5. y=T-x 6·y=T+T+4 Find the derivatives of all orders of the functions in Exercises 2!1- 32. 7. w = 3z-2 -} 8. = -21-' + ± 29. Y = X4 3 T - 2 x' - x 30. Y = x' 120 I' 9. y = 6.>:' - lOx - 5x-' 10. Y =4 - 2x - x-, 31. Y = (x - 1)(x' + 3x - 5) 32. y = (4x' + 3x)(2 - x) 11.r=-I--~ 12. r = 12 - ~ + ~4 3.' 2s 8 8' 8 Find the fITSt and second derivatives of the functions in Exercises · 33-40. In Exercises 13-16, find y' (a) by applying the Product Rule and =1'+51-1 33 y =x'+7 (b) by multiplying the factors to produce a sum of simpler terms to x 34. I , differentiate. (8 - 1)(8' + 8 + I) (x' + x)(x' - x + I) + I) = (2x + 3)(5x' 35. r = , 36.u= e;/Z)